Optimal. Leaf size=259 \[ \frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {b x^2+c x^4}}+\frac {3 x^{3/2} \left (b+c x^2\right )}{c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.23, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2022, 2032, 329, 305, 220, 1196} \[ \frac {3 x^{3/2} \left (b+c x^2\right )}{c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {b x^2+c x^4}}-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2022
Rule 2032
Rubi steps
\begin {align*} \int \frac {x^{11/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}}+\frac {3 \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{2 c}\\ &=-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}}+\frac {\left (3 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{2 c \sqrt {b x^2+c x^4}}\\ &=-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}}+\frac {\left (3 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{c \sqrt {b x^2+c x^4}}\\ &=-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}}+\frac {\left (3 \sqrt {b} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{c^{3/2} \sqrt {b x^2+c x^4}}-\frac {\left (3 \sqrt {b} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{c^{3/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {x^{5/2}}{c \sqrt {b x^2+c x^4}}+\frac {3 x^{3/2} \left (b+c x^2\right )}{c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {b x^2+c x^4}}+\frac {3 \sqrt [4]{b} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 c^{7/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.24 \[ -\frac {2 x^{5/2} \left (\sqrt {\frac {c x^2}{b}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {c x^2}{b}\right )-1\right )}{c \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} x^{\frac {3}{2}}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 200, normalized size = 0.77 \[ \frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+6 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) x^{\frac {5}{2}}}{2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{11/2}}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {11}{2}}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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